We note that the
pressure Schur complement preconditioner (REF )
was initially introduced for the generalized Stokes problem (constant
density, constant viscosity, and \(\gamma =0\) ) by
Cahouet and Chabard [1]}.
Robustness of this
Cahouet-Chabard preconditioner
for the generalized Stokes problem
with respect to variations in the mesh size
\(h\)  and time step size \(\delta t\)  was proven in [2]}, [3]}, [4]}.
It was then generalized by Olshanskii et al. [4]}
to the generalized Stokes interface problem
(variable density, variable viscosity, and \(\gamma =0\) ).
While a theoretical proof of the robustness of the preconditioner in [4]} for the variable density and viscosity
case was lacking due to the lack of regularity results for the stationary
Stokes interface problem, numerical results performed in
[4]} seems to indicate that the preconditioner is
robust also with respect to the jumps in viscosity and density
in large parameter ranges.
Hence, our preconditioner (REF ) can be considered as a
generalization of the one in [4]} to take into account the
structure compressibility (\(\gamma >0\)  on \(\Omega _s\) ) in the pressure block.
